The situation for the one-hole system, see Fig. REF b, is quite different. As a function of decreasing \(U^{-1}\)  (increasing \(U\) ), the spin spectrum gets linearly reduced, achieving a small value at \(U_c\) , and then in a cusp-like manner increasing again.
The quality of the linear fit is striking, and this makes it possible to locate the phase transition point \(U_c\)  with a few calculations in a \(U\)  regime away from the difficult \(U_c\)  point. At \(U_c\)  the width \(\Delta E\)  gets a minimum value, which is measured as \(<0.03\) , meaning that at the transition point the ground state energies of all the spin states are nearly degenerate to within this energy window. This implies that very large spin fluctuations can be expected in the vicinity of the Nagaoka transition. A further consequence of this massive near-degeneracy should be non-trivial behaviour of the entropy and heat capacity through this transition. This however would be best studied using a finite-temperature method [1]}, [2]} rather than a ground-state technique.
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